Two Handy Geometric Prediction Methods of Cancer Growth

Abstract

In present day societies, cancer is a widely spread disease that affects a large proportion of the human population, many research teams are developing algorithms to help medics to understand this disease. In particular, tumor growth has been studied from different viewpoints and different mathematical models have been proposed. Our aim is to make predictions about shape growth, where shapes are given as domains bounded by a closed curve in R 2 .These predictions are based on geometric properties of plane curves and vectors. We propose two methods of prediction and a comparison between them is shared. Both methods can be used to study the evolution in time of any 2D and 3D geometrical forms such as cancer skin and other types of cancer boundary. The first method is based on observations in the normal direction to the plane curve (boundary) at each point (normal method). The second method is based on observations at the growing boundaries in radial directions from the "center" of the shape (radius method). The real data consist of at least two input curves that bind a plane domain.

Mathematical Methods to Predict the Dynamic Shape Evolution of Cancer Growth based on Spatio-Temporal Bayesian and Geometrical Models

“…1. For every atom {ω} , 0 ≤ P ({ω}) ≤ 1 2. ω∈Ω P ({ω}) = 1 whenever (1) and (2) hold, P defines a consistent probability measure on Ω.…”

“…where the superscript in N [2] denotes the process of distinct pairs from N and in the second integral we have used the assumption of independent clusters. Taking expectations with respect to the cluster centre process, we obtain for the second factorial moment of the cluster process:…”

“…So as not to overload the picture, we plot just the first two iterations and the corresponding points of calculus P t 1 1 , P t 2 1 , and the predicted point P t 3 1 (see Figure 3.4a). Normal method: in order to begin the simulation, the first vector is the normal vector − −−−− → N 1 (P t 1 1 ) to surface α t in point P 1 (x 1 , y 1 ), and in that direction we find the point P j (x j , y j ) i.e P t 2 1 at the intersection with the second contour α t+∆t . Starting from this point in the direction of the normal vector to second contour, we calculate the spatial coordinates of the predicted point (Figure 3.4b).…”

“…This chapter is based on the published paper: "A geometric approach to cancer growth prediction based on Cox Processes" by Vlad and Mateu (2014)[1] 2 This chapter is based on the published paper: "Two handy geometric prediction methods of cancer growth" by[2] 3 This chapter is based on the published paper: "Bayesian spatio-temporal prediction of cancer dynamics" by[3] …”

“…This chapter is based on the published paper: "Two handy geometric prediction methods of cancer growth" by Vlad et al (2015)[2] …”